Complex Numbers to Complex Powers

Date: 10/19/2000 at 20:24:08
From: Michael Lee
Subject: Non-e base with imaginary exponent

I understand Euler's discovery of e^ix and its relation to the sine 
and cosine functions. But since e is just a number (although a special 
one), what is the general value of n^i? Specifically, since a real 
number to a real power equals a real, does a real number to a complex 
power equal some complex number? If so, what is it? 

Thank you.
Michael Lee

Date: 10/19/2000 at 22:47:37
From: Doctor Peterson
Subject: Re: Non-e base with imaginary exponent

Hi, Michael.

You're right; in fact, we can use the formula to raise any complex 
number to any complex power.

Any real number a can be written as e^ln(a); so

     a^(ix) = (e^ln(a))^(ix) 
            = e^(ix*ln(a)) 
            = cos(x*ln(a)) + i*sin(x*ln(a))

We can extend this to complex exponents this way:

     a^(x+iy) = a^x * a^(iy)

To allow for complex bases, write the base in the form a*e^(ib), and 
you find

     [a*e^(ib)]^z = a^z * e^(ib*z)

These ideas will allow you to raise any real or complex base to any 
real or complex exponent.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 10/20/2000 at 15:42:38
From: Michael Lee
Subject: Real raised by imaginary number

Doctor Peterson,

I asked what the result was of raising a real by a complex number. You 
sent me a great reply, which is understandable on an analytical level. 
However, when viewed geometrically, it seems that result of this 
operation maps onto a unit circle on the complex plane. 

This seems too fantastic. I've relied on intuition for my sense of 
mathematics, but this is beyond my senses. How is it possible that all 
combinations of real numbers raised to powers with only an imaginary 
component degenerate to the unit circle? 

If I had to make up an explanation I might say that in the world of 
numbers, reals cause translation and imaginaries cause rotation; also, 
since multiplying by i is often seen as rotating by 90 degrees in the 
complex plane, then by extension, raising by an imaginary is just a 
more complicated version of this rotation characteristic of i. Is 
this possible? 

Thank you.

Date: 10/20/2000 at 16:44:36
From: Doctor Peterson
Subject: Re: Real raised by imaginary number

Hi, Michael.

Yes, you're right: any real raised to any pure imaginary power gives a 
complex number with absolute value 1, so the mapping y -> e^(iy) takes 
the real line onto the unit circle, wrapping it around with a period 
of 2 pi. That's what Euler's formula means: raising e to an imaginary 
power produces the complex number with that angle.

You may want to look at our FAQ on this,
  
   Imaginary Exponents and Euler's Equation
   http://mathforum.org/dr.math/faq/faq.euler.equation.html   

When we raise a real to a complex power, the imaginary component of 
the exponent rotates it while the real component dilates (not 
translates) it. That is, the angle of the resulting number is 
determined by the imaginary part of the exponent, while the absolute 
value is determined by the real part. This is what gives Euler's 
formula tremendous power, and in fact this is the answer to the often- 
asked question, what good are complex numbers? They let us combine 
dilation and rotation, or exponential and sinusoidal functions, into a 
single operation.

Multiplication by a complex number, similarly, rotates by the angle of 
the complex number (adding the two angles), and dilates by its 
absolute value (multiplying the absolute values.)

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/